Date of Award

1-1-2019

Document Type

Dissertation

Degree Name

Ph.D.

Organizational Unit

College of Natual Science and Mathematics, Mathematics

First Advisor

Nikolaos Galatos, Ph.D.

Keywords

Algebraic logic, Decidability, Residuated lattice, Substructural logic, Undecidability

Abstract

We present a number of results related to the decidability and undecidability of various varieties of residuated lattices and their corresponding substructural logics. The context of this analysis is the extension of residuated lattices by various simple equations, dually, the extension of substructural logics by simple structural rules, with the aim of classifying simple equations by the decidability properties shared by their extensions. We also prove a number of relationships among simple extensions by showing the equational theory of their idempotent semiring reducts coincides with simple extensions of idempotent semirings. On the decidability front, we develop both semantical and syntactical methods for establishing decidability as well as tractability of decision procedures. On the undecidability front, we develop a notion of algebraic machines for which the theory of residuated frames will allow us to encode decision problems within the theories of residuated lattices and their substructural analogues. We prove the undecidability of the word problem for a broad class of simple extensions for both commutative and non-commutative residuated lattices. Furthermore, through a deduction theorem we establish the undecidability of the equational theory for a broad class of simple extensions. Translated in terms of substructural logics, we prove that the undecidability of both provability and deducibility for a multitude of extensions of FLe by simple rules.

Publication Statement

Copyright is held by the author. User is responsible for all copyright compliance.

Rights Holder

Gavin St. John

Provenance

Received from ProQuest

File Format

application/pdf

Language

en

File Size

165 p.

Discipline

Mathematics

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Algebra Commons

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